Problem 20 - Calculus for Biology and Medicine

Question

Use the dot product to compute the length of \([-1,4,3]\).

Step-by-Step Solution

Answer

The length of the vector \\([-1, 4, 3]\\) is \\(\sqrt{26}\\).

Step 1: Understanding the Dot Product Formula

The dot product of a vector with itself \([\mathbf{v} \cdot \mathbf{v}]\) is equal to the sum of the squares of its components. For a vector \([x, y, z]\), this is \([x^2 + y^2 + z^2]\). The magnitude (or length) of the vector \([||\mathbf{v}||]\) can be found by taking the square root of this dot product.

Step 2: Applying the Dot Product Formula

To find the dot product of \([-1, 4, 3]\) with itself, compute \([(-1)^2 + 4^2 + 3^2]\). Calculate each square: \([-1)^2 = 1\), \(4^2 = 16\), \(3^2 = 9\). Thus, \([-1, 4, 3] \cdot [-1, 4, 3] = 1 + 16 + 9\).

Step 3: Calculating the Dot Product

Add up the results from Step 2: \(1 + 16 + 9 = 26\). This is the dot product of the vector \([-1, 4, 3]\) with itself.

Step 4: Calculating the Magnitude

The magnitude of the vector is the square root of the dot product calculated in Step 3. Therefore, calculate \(\sqrt{26}\).

Step 5: Final Result

The length of the vector \([-1, 4, 3]\) is \(\sqrt{26}\). This is an exact value and is the final answer to the exercise.

Key Concepts

Understanding the Dot Product

The dot product is a fundamental concept in vector mathematics. It measures how much two vectors "point" in the same direction. For any vector \( \mathbf{v} = [x, y, z] \), the dot product \( \mathbf{v} \cdot \mathbf{v} \) is calculated by squaring each component of the vector and then summing the results.

• It is calculated as \((x^2 + y^2 + z^2)\).
• This showcases that the dot product is closely related to the length of the vector.

For example, if we consider the vector \([-1, 4, 3]\), its dot product with itself is calculated as \((-1)^2 + 4^2 + 3^2\), leading to \(1 + 16 + 9 = 26\). This result not only gives us a portion of the magnitude calculation but also helps to show the vector's properties with its direction. When two vectors are perpendicular, their dot product will equal zero. Therefore, this calculation is not only useful in measuring length but also in understanding relationships between vectors.

Exploring Vector Components

Vectors are composed of components that determine their direction and length in space. For a 3-dimensional vector \([x, y, z]\), each component \(x, y,\) and \(z\) represents a coordinate in its respective axis—x-axis, y-axis, and z-axis. Understanding each component is crucial for grasping how vectors work and interact.

• The components give us insight into the vector's direction.
• A change in any component significantly alters the vector's properties.

Let's revisit the vector \([-1, 4, 3]\):

• The first component \(-1\) determines the direction along the x-axis.
• The second component \(4\) dictates movement along the y-axis.
• The third component \(3\) influences the z-axis.

Each component contributes differently, and their collective effect determines the vector's net direction. For computational purposes, these components are squared and summed in steps like calculating the dot product or the vector's magnitude.

Calculating Magnitude

Calculating the magnitude of a vector is essential for understanding its size or length. It is essentially the vector's length in the multi-dimensional space it occupies. To calculate the magnitude of a vector \([x, y, z]\), lay the groundwork with its dot product \((x^2 + y^2 + z^2)\). The magnitude \(||\mathbf{v}||\) is the square root of that dot product:

• Magnitude = \(\sqrt{x^2 + y^2 + z^2}\)
• The square root of the dot product gives the actual length of the vector.

Using the dot product result from earlier with vector \([-1, 4, 3]\), the magnitude comes out to be \(\sqrt{26}\). This process of magnitude calculation not only provides the length but also enhances our understanding of the vector’s scale in its space.Moreover, knowing the magnitude is crucial for applications like normalizing vectors, which involves rescaling a vector to make its length equal to one while retaining its direction. This is key in areas like computer graphics and physics, where precise measurements are important.